Credit-Equity Modeling under a Latent Lévy Firm Process

Transcription

1 .... Credit-Equity Modeling under a Latent Lévy Firm Process Masaaki Kijima a Chi Chung Siu b a Graduate School of Social Sciences, Tokyo Metropolitan University b University of Technology, Sydney September 27, 2013 Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 1 / 1

2 Plan of My Talk..1 Introduction: Motivation, Literature Review Equity Options, Credit Modeling, Credit-Equity Models CreditGrades, Latent Credit Model..2 Our Model Firm Value, Equity Value Regime-Switching (mid-term spread), Jumps (short-term spread) CDS, Equity Option Pricing..3 Numerical Examples CDS, Equity Option..4 Conclusion, Future Research Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 1 / 1

3 Motivation Recent credit crisis shows the intimate relationship between the credit and equity markets. For example, during the credit crisis, both CDS premiums and equity volatilities were at their historical high. However, until recently, the equity and credit modelings are two separate themes in the finance literature. The difficulty to construct the credit-equity model stems from the fact that the debt and equity possess different properties. Hence, new attempts are required to construct the credit-equity modeling in a unified manner. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 2 / 1

4 Literature Review: Equity Options Mostly, based on Stochastic Differential Equations (SDEs) Diffusion: Black Scholes Model, Local Volatility Model Stochastic Volatility (SV): Heston (1993), etc. Jump Diffusion: Merton (1976), Kou (2002), etc. Lévy Process: Ask Professor Vostrikova Regime Switching: Kijima and Yoshida (1993), Buffington and Elliott (2002), etc. Strength: Highly liquid markets (equity and equity options) Shortcoming : No mention about the issuer s (firm) credit exposure Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 3 / 1

5 Literature Review: Credit Modeling..1 Reduced-form approach Jarrow and Turnbull (1995), Madan and Unal (1998a), Duffie and Singleton (1999), etc. Strength: Analytical tractability and ability of generating a flexible and realistic term structure of credit spreads Shortcoming 1: Exogenous hazard rate process Shortcoming 2: Default mechanism is not related to the firm value..2 Structural approach Merton (1974), Black and Cox (1976), Leland (1994), etc. Strength: Economic appeal links firm value with debt and equity Shortcoming 1: Difficult to incorporate more realistic features without sacrificing tractability Shortcoming 2: Difficult to price equity or equity options Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 4 / 1

6 Literature Review: Credit-Equity Models..1 Reduced-form approach Carr and Wu (2010), Mendoza-Arriaga et al. (2010) and references therein..2 Structural approach CreditGrades model by Finger et al. (2002) and its extensions by Sepp (2006) for double-exponential jump-diffusion model by Ozeki et al. (2011) for general spectrally-negative Lévy process Time-changed Brownian motion approach by Hurd and Zhou (2011) Latent model by Kijima et al. (2009) Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 5 / 1

7 Literature Review: CreditGrades..1 Ordinary structural approach Consider a corporate firm that issues a debt and an equity. Let D and S be the debt and equity values per share, respectively. Let V be the firm value per share, so that V = D + S by the basic accounting assumption. V is modeled by a SDE and the default occurs when V reaches a default threshold. D and S are evaluated as contingent claims written on V...2 CreditGrades model by Finger et al. (2002) D is the discounted face value of debt and S is modeled by a GBM. V is given by V = D + S, and default is the first passage time of V. Strength: Easy to implement and extend. Shortcoming 1: D is irrelevant to the credit structure. Shortcoming 2: Credit quality is essentially equal to the equity value. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 6 / 1

8 Literature Review: Latent Credit Model..1 Latent model (in, e.g., medical science) Introduce the notion of the marker process that is observable and correlated to the actual status process (unobservable)...2 Latent structural model by Kijima et al. (2009) The actual firm status is latent. Debt value is given in terms of the actual firm status. Equity value is obtained as a residual value as in Merton (1974). Strength: Economically appealing Shortcoming 1: The equity has a maturity as in Merton (1974). Shortcoming 1: The pricing of equity options is very complicated. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 7 / 1

9 Our Model: Overview Structural approach: treat the firm value as a latent variable Extension of Kijima et al. (2009) to include jumps (for short-term credit spread) and regime switch (for mid-term spread) Source of information: Equity value Objective: Price CDS and Equity Option with default feature under a joint framework. Contributions: Our model.1. Introduces the credit status of the firm into the equity process...2. Serves as a theoretical support to the existing empirical analyses on the explanatory power of equity s historical and option-implied volatilities to the CDS spread variation (work in progress)...3. Has a flexibility in explaining both the short-term and mid-term behaviors of the credit spread and implied volatility curves. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 8 / 1

10 Our Model: Firm value process A t : Actual firm value at time t where A t = exp(x t ), t 0 A t is latent, i.e. unobservable and non-tradable. Nature of default: Default epoch τ is defined by τ = inf{t 0 : A t Γ} = inf{t 0 : X t L} for some Γ = e L (default barrier). Remark: Easy to extend to include a stochastic boundary. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 9 / 1

11 Our Model: Equity value process S t : Equity value of the firm at time t S t is observable to investors and tradable. Let Y t = log S t, and assume that (for each regime) Y t = ρx t + Z t Z t : Non-firm specific shocks, independent of X t (given each regime). ρ: The impact factor of firm s credit exposure on equity Remark: The model does not satisfy the basic accounting condition. It can be considered as an extension of CreditGrades by taking ρ = 1 and Z t = e rt F. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 10 / 1

12 Regime-Switching Introduce the regime-switching for the mid-term spread. Let {J t : t 0} be a Markov chain on state space E.. E is finite and contains d elements, i.e., E = {1, 2,..., d}. Let Q be the intensity matrix of J t with respect to the Lebesgue measure, i.e., Q = {q ij } i,j E where q ii = i j q ij Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 11 / 1

13 Model of Log-Firm Value Let X t = log A t be defined by X t = t 0 b X (J s )ds + + j E t 0 t 0 σ X (J s )dw X s 1 {Js=j}dN X s (j) where, given J t = j E, b X (J t ) b X j is a drift, σ X (J t ) σj X is a volatility, and {Nt X (j) : t 0} is a compound Poisson process. Nt X(j) has arrival rate λx j and double-exponential jumps Yj X with distribution νj X (dy), where [ νj X (dy) = λx j p X j ηx j1 e ηx j1 y 1 {y 0} + (1 p X j )ηx j2 eηx j2 y 1 {y<0} ]dy Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 12 / 1

14 Moment Generating Function of X t The moment generating function (MGF) of X t, E[exp(uX t )], is obtained by Kijima and Siu (2013) as E[exp(uX t )] exp ( K X [u]t ) where K X [u] {κ X j (u)} diag + Q with κ X j (u) = bx j u + (σx j u)2 2 + λ X j ( p X j ηj1 X ηj1 X u + (1 ) px j )ηx j2 ηj2 X + u 1 for regime-switching, double-exponential jumps. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 13 / 1

15 Model of Non-Firm Specific Shock Recall that Y t = log S t and, for each regime, Y t = ρx t + Z t. We assume that Z t has the following canonical representation: Z t = t 0 b Z (J s )ds + + t 0 R t 0 σ Z (J s )dw Z s y(µ Z (J s ) ν Z (J s ))(dy)ds where, for J t = j, b Z (J t ) b Z j is a drift, σz (J t ) σj Z is a volatility, µ Z (J t ) µ Z j is a random jump measure, and ν Z (J t ) νj Z is the compensator of µ Z j. Z t can be a general Lévy, because it is irrelevant to default. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 14 / 1

16 Moment Generating Function of Z t The MGF E[exp(uZ t )] is given by E[exp(uZ t )] exp ( K Z [u]t ) where K Z [u] {κ Z j (u)} diag + Q with κ Z j (u) = bz j u (σz j u)2 + (e uy 1 y1 { y 1} )νj Z (dy) R Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 15 / 1

17 No-Arbitrage Condition Assume J t = j. The discounted process S t e rt S t is a P-martingale with respect to F t if and only if ρb X j + b Z j = r 1 2 (ρσx j )2 1 2 (σz j )2 ( p X λ X j η j1 j η j1 ρ + (1 ) px j )ηx j2 ηj2 X + ρ 1 ( p Z λ Z j η j1 j η j1 1 + (1 ) pz j )ηz j2 ηj2 Z where r > 0 is the risk-free interest rate. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 16 / 1

18 Credit Default Swap Standard CDS premium formula: For J 0 = i E, T 0 T = (1 R) e rt dp i (τ t) T 0 e rt P i (τ > t)dt [ E i e rτ ] 1 {τ <T } = (1 R)r 1 E i [e rτ 1 {τ <T } ] e rt P i (τ > T ) c (i) where R is the recovery rate and r is the risk-free interest rate. Hence, we need to evaluate E i [ e rτ 1 {τ <T } ], i E. Following a similar discussion to Kijima and Siu (2013), these values are obtained as a solution of a linear equation, when jumps are double-exponential. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 17 / 1

19 Short-Term Credit Spread. Lemma.. Denote x = log( L A 0 ) and J 0 = i. Then, lim c (i) T = r(1 R)νX T 0 i ((, x]). where νi X denotes the Lévy measure under regime i... Implication: Regime-switching Brownian motion alone CANNOT produce non-zero credit spread at T 0! We need jumps for plausible short-term spreads. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 18 / 1

20 Long-Term Credit Spread. Lemma.. Assume P[τ < ] = 1 and J 0 = i. Then, lim T c(i) T = (1 R)r E Π [e rτ ] 1 E Π [e rτ ],. where Π denotes the stationary distribution of J t... Implication: Impact of the regime-switching factor appears in the medium part of the CDS term structure! Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 19 / 1

21 CDS Premium. Corollary.. In our model, the CDS premium c is given by where J 0 = i, T = (1 R)r P2 RS 1 P2 RS e rt P1 RS c (i) ( 1 P1 RS = L 1 T a 1 ) a E i[e aτ ; J τ ] and ( ) 1 P2 RS = L 1 T a E i[e (r+a)τ ; J τ ] Here,. L 1 denotes the inverse Laplace transform... Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 20 / 1

22 Numerical Results Model Parameters for X t : Base Parameters A 0 T r L R Regime 1 b X 1 σ1 X η11 X η12 X p X 1 λ X 1 q Regime 2 b X 2 σ2 X η21 X η22 X p X 2 λ X 2 q Regime 1 (Regime 2, resp.) is of high (low) volatility and bigger (smaller) jumps. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 21 / 1

23 Regime-Switching Factor: BM only CDS premium (bp) Effect of regime switching on CDS premium: BM case Regime 1: σ X 1 =0.4 Regime 2: σ X 2 =0.2 No Regime: σ X 1 =0.4 No Regime: σ X 2 = time to maturity (year) Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 22 / 1

24 Regime-Switching, Jump-Diffusion Effect of regime switching on CDS premium, Markov modulated Levy measure only 300 Regime 1: Big jumps Regime 2: Small jumps Big jumps (no regime) 250 Small jumps (no regime) CDS premium (bp) time to maturity (year) Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 23 / 1

25 Effect of Regime-Switching Intensity: q 2 = Effect of regime switching intensity 300 CDS premium (bp) q =0.5, q = q 1 =2, q 2 =0.5 q 1 =5, q 2 =0.5 q =10, q = time to maturity (year) Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 24 / 1

26 Summary of Numerical Examples Hump and inverted-hump shapes of the CDS curves can be constructed by changing the regime-switching intensities of J t. Possible explanation: When buying CDS, investors are concerned with (1) current state of the firm, and (2) persistence of a firm staying in one particular economic/credit regime. Short-term spreads become more realistic by the jump effects. Introduction of regime-switching, jump-diffusion results in more flexible CDS term structures! Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 25 / 1

27 Equity Option Recall that S t = exp(y t ) with Y t = ρx t + Z t The call option price written on S under {τ > T } is given by C(S, K, T ) = E[e rt (S T K) + 1 {τ >T } ] Hence, equivalently, = E[e rt (S T K) + ] E [ e rt (S T K) + 1 {τ T } ] Defaultable call = Non-defaultable call Down-and-in call Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 26 / 1

28 Equity Option Price. Theorem.. The double Laplace transform of E[e rt (S T K) + 1 {τ T } ] with respect to k = log K and T is obtained as L ξ,β (E[e rt (S T K) + 1 {τ T } ]) S ξ+1 ] 0 = Ẽ i [e ((β+r) κz j (ξ+1))τ +(ξ+1)ρx τ 1 {Jτ =j} ξ(ξ + 1) j ( ( { )) 1 (r + β)i κ Z j (ξ + 1) + κx j }diag (ρ(ξ + 1)) + Q n jn. where Ẽ i is the expectation under which Z t is taken as the numeraire... Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 27 / 1

29 Model Parameters for X t : Numerical Results Regime 1 Base Parameters S 0 K A 0 T r L b X 1 σ1 X η11 X η12 X p X 1 λ X 1 q Regime 2 b X 2 σ2 X η21 X η22 X p X 2 λ X 2 q Model Parameters for Z t (double-exponential for simplicity): Regime 1 σ1 Z η11 Z η12 Z p Z 1 λ Z Regime 2 σ2 Z η21 Z η22 Z p Z 2 λ Z Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 28 / 1

30 implied volatility Impact of Jump Factor Regime-switching BM produces symmetric smiles. Negative skewness is a common feature found in equity markets. The negative skewness is more pronounced as the probability of upper jumps p X i decreases, since the probability of default is decreased. Regime 1 (Regime 2, resp.) is of high (low) volatility and bigger (smaller) jumps. 0.4 The effect of p X on implied volatility, Regime 1 1 p X 1 =0 p X 1 =0.4 p X 1 =0.6 p X 1 =0.8 BM only implied volatility The effect of p X on implied volatility, Regime 2 2 p X 2 =0 p X 2 =0.4 p X 2 =0.6 p X 2 =0.8 BM only log(moneyness) log(moneyness) Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 29 / 1

31 Effect of ρ without Regime-Switch The curvature of IV curve decreases with increasing correlation ρ. That is, the increase in ρ augments the negative skewness of IV. The negative skewness reflects the credit nature on the equity Correlation effect on implied volatility, ρ=0.3,t=1 ρ= Effect of correlation on implied volatility, T=1 ρ= implied volatility implied volatility log(moneyness) log(moneyness) Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 30 / 1

32 Time Effect Volatility curves flatten with increasing maturity T. In Regime 1, the IV curve moves downward as it flattens, whereas it elevates as its curvature decreases in Regime 2. Of course, they converge to coincide as T Effect of time to maturity on implied volatility, ρ=1, J 0 =1 T=0.5 T=1 T=2 T= Effect of time to maturity on implied volatility, ρ=1, J 0 =2 T=0.5 T=1 T=2 T=3 implied volatility implied volatility log(moneyness) log(moneyness) Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 31 / 1

33 Summary of Numerical Examples Regime-switching and jump factors play a significant role in the equity option with default feature. In particular, the IV curve of the high regime decreases, while it increases under the low regime, as the switching-intensity or the maturity lengthens. The default probability contributes to the negative skewness of IV. However, the degree of negative skewness is limited, in comparison with the reduced-form credit-equity model in Carr and Wu (2010). The assumption of independent and stationary increments of Lévy processes makes it inflexible in capturing the IV observed in the market (see Konikov and Madan, 2002). Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 32 / 1

34 Conclusion Increasing evidence of the linkage between the equity and credit aspects of a corporate firm demands a unified equity-credit model. We propose one approach to the problem: Latent structural model. Extend Kijima et al. (2009) to include jumps and regime-switch. Application: Price CDS and equity option under one framework. Strength: Separate jumps and regime-switch effects. Strength: More flexible CDS term structures and IV surfaces. Strength: Clarify the role of impact factor ρ to the skewness of volatility smiles. Numerical scheme: Inverse Laplace transform is very easy and stable. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 33 / 1

35 Future Research Need to develop a calibration scheme. Want to extract credit quality (e.g., distance to default) under the physical measure from the marker process (i.e., equity value process). These can lead to more empirical works. Extend the model to include the Heston-type SV (to increase the negative skewness). The pricing of equity default swap, which has both the equity and credit components of a firm. Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 34 / 1

36 Thank You for Your Attention Kijima and Siu (TMU) Credit-Equity Modeling 27/09/13 35 / 1